where the characteristic function \(v_c\)  for data valuation is previously defined in Sec. , \(\Pi _N\)  is the set of all possible permutations of \(N\) , and
\(C_{\pi ,i}\)  is the coalition of parties preceding \(i\)  in permutation \(\pi \)  [1]}.
The notion of marginal contribution (and hence Shapley value) plays a significant role in the properties of (F3) strict desirability and (F4) strict monotonicity
that define the (R5) fairness incentive in [2]}:The other two properties: (F1) uselessness and (F2) symmetry defining R5 in [2]} are standard axioms of Shapley value [4]} and commonly used in works on data valuation [5]}, [6]}, [7]}.
Due to lack of space, we have reproduced the formal definitions of properties F1 to F4 in Appendix .
In our work, the implication of F3 is that if the marginal contributions of parties \(i\)  and \(j\)  only differ for coalition \(C\)  (i.e., \(v_c(C\cup \lbrace i\rbrace ) > v_c(C\cup \lbrace j\rbrace )\) ), then it is only fair for party \(i\)  to be assigned a larger reward value \(r_i\) ; its effect on our modified F4 will be discussed later in Sec. REF .
